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Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault cohomology, just by putting $\mathcal{F}=\mathbb{R}$ or the sheaf of holomorphic functions $\Omega^p(M)$. But each generalisation involves losing intuition about the measured geometric properties.

My question is: is it possible to understand the geometric intuition behind sheaf and Cech cohomology in the same way one can understand the geometry behind de Rham cohomology?

See also this related question.

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Jjm
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    Well do you know the definition of Čech cohomology? You build it up from small open sets (so small that you only see local properties), then you look at how it relates to intersections with other small open sets, then triple intersections... If you've never seen the proof (or at least the statement) that a space is uniquely determined up to homotopy by a good cover (seen as a lattice), that's very enlightening. – Najib Idrissi Feb 18 '15 at 16:08
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    @NajibIdrissi Yes, it is very helpful. The critical point is the following one: if I worked with $\mathcal{F}=\mathbb{R}$, then it is very easy to 'match' Cech cohomology with singular cohomology in triangulated spaces, in the spirit of the introduction of the third chapter of Hatcher's Algebraic Topology (pages 185-189). But when working with a nontrivial sheaf, the comparison gets messy and confuse, and does not enlight any more. What do you think about it? – Jjm Feb 18 '15 at 16:21

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